This doesn't need much analysis. We've seen this in many times before.
Husband was dealt a tough pair of cards. A certain divorce, which he didn't want (they had kids). You could argue it was beta to accept the demand. I think it was alpha to do what he felt was best for his kids and start sleeping with another woman.
Wife described her husbands 'affair' as 'cheating' even though opening up the marriage was her demand.
Wife had someone in mind when she made this demand for an open relationship. They always do. They do not bring up this subject (which has risk) if they are not pretty much ready to go with a guy in 24 hours. Branch swinging requires another branch.
Wife's affair partner really was an affair, her partner had a wife who didn't know. But husband did it by the book with a single woman. Stand up guy.
Wife then threatens divorce if husband doesn't close the relationship. When someone has used the nuclear bargaining chip of divroce once, expect them to use it again. The cat is out of the bag.
Husband offered a fair legal arrangement that if he closes the relationship then there are consequences if the wife or husband commits infidelity again. Wife goes beserk. Refuses to sign it. I believe that this shows the lesson the wife learned was that she should have cheated in secret, not asked for an open marraige. She wanted to sleep with other men, and didn't want her husband to sleep with other women. And the easiest way to achieve that was to close the relationship and just cheat.
The wife said the husband had to end his relationship with Amber because the wife knew her. Well it turns out that the wife knows her because she used a tracking device, then drove to Amber and introduced herself. Lmao.
edit: looking at the replies and they are pathetic.
TL;DR (even with update). OP sucks for trying to one-up his wife in every situation rather than work as a team, also misses the point of the rules they had; wife sucks extra for obvious reasons. They both suck and theyd rather fight this than be good parents to their kids.